Fixation and stationary times for the $\Lambda$-Wright-Fisher process

Airam Blancas; Adrián González Casanova; Sebastian Hummel; Sandra Palau

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arXiv · 2308.09218

Fixation and stationary times for the $\Lambda$-Wright-Fisher process

Airam Blancas·Adrián González Casanova·Sebastian Hummel·Sandra Palau

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Abstract

We study the fixation and stationary behavior of the Lambda-Wright-Fisher process with parent-independent mutation and finitely many types, a jump-diffusion model for allele frequency dynamics in large populations with potentially large offspring variance. Using a lookdown construction, we characterize the distribution of fixation times and the order of allele extinctions in the absence of mutations, and identify a strong stationary time in the presence of mutations. Our results include explicit expressions for the mean fixation and stationary times for the Wright-Fisher diffusion, and mean fixation times in the Beta-coalescent case. A key component of our approach is the analysis of the fixation line introduced by H\'enard in (Ann. Appl. Probab., 25:3007-3032, 2015). We extend this process to incorporate mutation, providing a unified framework for studying both fixation and equilibrium behavior.

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Airam Blancas, Adrián González Casanova, Sebastian Hummel, Sandra Palau. 2023-08-18. Fixation and stationary times for the $\Lambda$-Wright-Fisher process. https://arxiv.org/abs/2308.09218

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